nips nips2007 nips2007-214 knowledge-graph by maker-knowledge-mining

214 nips-2007-Variational inference for Markov jump processes

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Author: Manfred Opper, Guido Sanguinetti

Abstract: Markov jump processes play an important role in a large number of application domains. However, realistic systems are analytically intractable and they have traditionally been analysed using simulation based techniques, which do not provide a framework for statistical inference. We propose a mean ﬁeld approximation to perform posterior inference and parameter estimation. The approximation allows a practical solution to the inference problem, while still retaining a good degree of accuracy. We illustrate our approach on two biologically motivated systems.

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Variational inference for Markov jump processes Guido Sanguinetti Department of Computer Science University of Shefﬁeld, U. [sent-1, score-0.205]

2 de Abstract Markov jump processes play an important role in a large number of application domains. [sent-8, score-0.145]

3 We propose a mean ﬁeld approximation to perform posterior inference and parameter estimation. [sent-10, score-0.172]

4 Introduction Markov jump processes (MJPs) underpin our understanding of many important systems in science and technology. [sent-13, score-0.145]

5 They provide a rigorous probabilistic framework to model the joint dynamics of groups (species) of interacting individuals, with applications ranging from information packets in a telecommunications network to epidemiology and population levels in the environment. [sent-14, score-0.093]

6 A traditional approach, which has been very successful throughout the past century, is to ignore the discrete nature of the processes and to approximate the stochastic process with a deterministic process whose behaviour is described by a system of non-linear, coupled ODEs. [sent-17, score-0.493]

7 This approximation relies on the stochastic ﬂuctuations being negligible compared to the average population counts. [sent-18, score-0.194]

8 Researchers are now able to obtain accurate estimates of the number of macromolecules of a certain species within a cell [2, 3], prompting a need for practical statistical tools to handle discrete data. [sent-20, score-0.4]

9 However, it is not clear how observed data can be incorporated in a principled way, which renders this approach of limited use for posterior inference and parameter estimation. [sent-24, score-0.14]

10 In this paper we present an alternative, deterministic approach to posterior inference and parameter estimation in MJPs. [sent-29, score-0.17]

11 In this way, we replace the couplings between the 1 different species by their average, mean-ﬁeld (MF) effect. [sent-34, score-0.352]

12 The rest of this paper is organised as follows: in sections 1 and 2 we review the theory of Markov jump processes and introduce our general strategy to obtain a MF approximation. [sent-36, score-0.145]

13 In section 4 we present experimental results on simulated data from the Lotka-Volterra model and from a simple gene regulatory network. [sent-38, score-0.097]

14 Finally, we discuss the relationship of our study to other stochastic models, as well as further extensions and developments of our approach. [sent-39, score-0.112]

15 1 Markov jump processes We start off by establishing some notation and basic deﬁnitions. [sent-40, score-0.145]

16 A D-dimensional discrete stochastic process is a family of D-dimensional discrete random variables x(t) indexed by the continuous time t. [sent-41, score-0.269]

17 The dimensionality D represents the number of (molecular) species present in the system; the 0 components of the vector x (t) then represent the number of individuals of each species present at time t. [sent-43, score-0.782]

18 Furthermore, the stochastic processes we will consider will always be Markovian, i. [sent-44, score-0.124]

19 given any sequence of observations for the state of the system (xt1 , . [sent-46, score-0.109]

20 A discrete stochastic process which exhibits the Markov property is called a Markov jump process (MJP). [sent-50, score-0.415]

21 A MJP is characterised by its process rates f (x |x), deﬁned ∀x = x; in an inﬁnitesimal time interval δt, the quantity f (x |x) δt represents the inﬁnitesimal probability that the system will make a transition from state x at time t to state x at time t + δt. [sent-51, score-0.301]

22 The interpretation of the process rates as inﬁnitesimal transition probabilities highlights the simple relationship between the marginal distribution pt (x) and the process rates. [sent-54, score-0.414]

23 In formulae, this is given by   pt+δt (x) = pt (x) 1 − x =x f (x |x) δt + pt (x ) f (x|x ) δt. [sent-56, score-0.156]

24 x =x Taking the limit for δt → 0 we obtain the (forward) Master equation for the marginal probabilities dpt (x) = dt 2 [−pt (x) f (x |x) + pt (x ) f (x|x )] . [sent-57, score-0.275]

25 (2) x =x Variational approximate inference Let us assume that we have noisy observations yl l = 1, . [sent-58, score-0.209]

26 , N of the state of the system at a discrete number of time points; the noise model is speciﬁed by a likelihood function p (y l |x (tl )). [sent-61, score-0.127]

27 We ˆ can combine this likelihood with the prior process to obtain a posterior process. [sent-62, score-0.184]

28 As the observations happen at discrete time points, the posterior process is clearly still a Markov jump process. [sent-63, score-0.352]

29 While this is possible in principle, the computations would require simultaneously solving a very large system of coupled linear ODEs (the number of equations is of order S D , S being the number of states accessible to the system), which is not feasible even in simple systems. [sent-65, score-0.131]

30 2 In the following, we will use the variational mean ﬁeld (MF) approach to approximate the posterior process by a factorizing process, minimising the Kullback - Leibler (KL) divergence between processes. [sent-66, score-0.343]

31 The inference process is then reduced to the solution of D one - dimensional Master and backward equations of size S. [sent-67, score-0.212]

32 This is still nontrivial because the KL divergence requires the joint probabilities of variables x(t) at inﬁnitely many different times t, i. [sent-68, score-0.094]

33 probabilities over entire paths of a process rather than the simpler marginals pt (x). [sent-70, score-0.258]

34 Hence, we write the joint posterior probability as ppost (x0:K ) = 1 pprior (x0:K ) × Z K−1 N p (yl |x (tl )) ˆ with pprior (x0:K ) = p(x0 ) p(xk+1 |xk ) k=0 l=1 with Z = p(y1 , . [sent-76, score-0.373]

35 In the rest of this section, we will show how to compute the posterior rates and marginals by minimising the KL divergence. [sent-81, score-0.197]

36 We notice in passing that a similar framework for continuous stochastic processes was proposed recently in [8]. [sent-82, score-0.124]

37 1 KL divergence between MJPs The KL divergence between two MJPs deﬁned by their path probabilities p(x 0:K ) and q(x0:K ) is KL [q, p] = q(x0:K ) ln x0:K q(x0:K ) = p(x0:K ) K−1 q(xk ) k=0 xk q(xk+1 |xk ) ln xk+1 q(xk+1 |xk ) + K0 p(xk+1 |xk ) will be set to zero in the following. [sent-84, score-0.462]

38 Notice that we have swapped from the path probabilities framework to an expression that depends solely on the process rates and marginals. [sent-86, score-0.282]

39 2 MF approximation to posterior MJPs We will now consider the case where p is a posterior MJP and q is an approximating process. [sent-88, score-0.245]

40 The prior process will be denoted as pprior and its rates will be denoted by f . [sent-89, score-0.279]

41 The KL divergence then is N KL(q, ppost ) = ln Z + KL(q, pprior ) − Eq [ln p (yl |x (tl ))] . [sent-90, score-0.344]

42 ˆ l=1 To obtain a tractable inference problem, we will assume that, in the approximating process q, the joint path probability for all the species factorises into the product of path probabilities for individual species. [sent-91, score-0.757]

43 This gives the following equations for the species probabilities and transition rates D qt (x) = D qit (xi ) gt (x |x) = i=1 δxj ,xj git (xi |xi ) . [sent-92, score-1.097]

44 (4) i=1 j=i Notice that we have emphasised that the process rates for the approximating process may depend explicitly on time, even if the process rates of the original process do not. [sent-93, score-0.639]

45 In order to ﬁnd the MF approximation to the posterior process we must optimise the KL divergence (5) with respect to the marginals qit (x) and the rates git (x |x). [sent-98, score-0.918]

46 We will take care of this constraint by using a Lagrange multiplier function λi (x, t) and compute the stationary values of the Lagrangian L =KL (q, ppost )  T − dt i 0 x λi (x, t) ∂t qit (x) − x =x  {git (x|x ) qit (x ) − git (x |x) qit (x)} . [sent-100, score-1.145]

47 Then ˆ ˆ equation (8) yields lim ri (x, t) = pi (yil |xi (tl )) lim ri (x, t) . [sent-103, score-0.171]

48 Starting with an initial guess for ˆ ˜ qt (x) and selecting a species i, we can compute fi (x |x) and fi (x |x). [sent-105, score-0.669]

49 Using these, we can solve equation (10) backwards starting from the condition ri (x, T ) = 1∀x (i. [sent-106, score-0.111]

50 This allows us to update our estimate of the rates git (x |x) using equation (9), which can then be used to solve the master equation (2) and update our guess of qit (x). [sent-109, score-0.797]

51 3 Parameter estimation Since KL [q, ppost ] ≥ 0, we obtain as useful by-product of the MF approximation a tractable variational lower bound on the log - likelihood of the data log Z = log p(y1 , . [sent-112, score-0.221]

52 g 7] such a bound can be used in order to optimise model parameters using a variational E-M algorithm. [sent-117, score-0.121]

53 4 3 Example: the Lotka-Volterra process The Lotka-Volterra (LV) process is often used as perhaps the simplest non-trivial MJP [6, 4]. [sent-118, score-0.208]

54 Lotka in 1925 and Vito Volterra in 1926, it describe the dynamics of a population composed of two interacting species, traditionally referred to as predator and prey. [sent-120, score-0.183]

55 The process rates for the LV system are given by fprey (x + 1|x, y) = αx fpredator (y + 1|x, y) = δxy fprey (x − 1|x, y) = βxy fprey (y − 1|x, y) = γy (11) where x is the number of preys and y is the number of predators. [sent-121, score-0.439]

56 All other rates are zero: individuals can only be created or destroyed one at the time. [sent-122, score-0.163]

57 Rate sparsity is a characteristic of very many processes, including all chemical kinetic processes (indeed, the LV model can be interpreted as a chemical kinetic model). [sent-123, score-0.353]

58 An immediate difﬁculty in implementing our strategy is that some of the process rates are identically zero when one of the species is extinct (i. [sent-124, score-0.586]

59 its numbers have reached zero); this will lead to inﬁnities when computing the expectation of the logarithm of the rates in equation (6). [sent-126, score-0.172]

60 To avoid this, we will “regularise” the process by adding a small constant to the f (1|0); it can be proved that on average over the data generating process the variational approximation to the regularised process still optimises a bound analogous to (3) on the original process [9]. [sent-127, score-0.524]

61 The variational estimates for the parameters of the LV process are obtained by inserting the process rates (11) into the MF bound and taking derivatives w. [sent-128, score-0.401]

62 (12) Equations (12) have an appealing intuitive meaning in terms of the physics of the process: for example, α is given by the average total increase rate of the approximating process divided by the average total number of preys. [sent-133, score-0.157]

63 We generated 15 counts of predator and prey numbers at regular intervals from a LV process with parameters α = 5 × 10−4 , β = 1 × 10−4 , γ = 5 × 10−4 and δ = 1 × 10−4 , starting from initial population levels of seven predators and nineteen preys. [sent-134, score-0.475]

64 These counts were then corrupted according to the following noise model pi (yil |xi (tl )) ∝ ˆ 1 + 10−6 , 2|yil −xi (tl )| (13) where xi (tl ) is the (discrete) count for species i at time tl before the addition of noise. [sent-135, score-0.672]

65 Notice that, since population numbers are constrained to be positive, the noise model is not symmetric. [sent-136, score-0.129]

66 While in theory each species can have an arbitrarily large number of individuals, in order to solve the differential equations (2) and (10) we have to truncate the process. [sent-139, score-0.431]

67 While the truncation threshold could be viewed as another parameter and optimised variationally, in these experiments we took a more heuristic approach and limited the maximum number of individuals of each species to 200. [sent-140, score-0.498]

68 This was justiﬁed by considering that an exponential growth pattern ﬁtted to the available data led to an estimate of approximately 90 individuals in the most abundant species, well below the truncation threshold. [sent-141, score-0.11]

69 The solid line is the mean of the approximating distribution, the dashed lines are the 90% conﬁdence intervals, the dotted line is the true path from which the data was obtained. [sent-143, score-0.103]

70 32 × 10−4 , 5 25 25 y x 20 20 15 15 10 10 5 5 0 0 500 1000 1500 2000 2500 0 0 3000 t 500 1000 1500 2000 2500 3000 t (a) (b) Figure 1: MF approximation to posterior LV process: (a) predator population and (b) prey population. [sent-147, score-0.363]

71 While the process is well approximated in the area where data is present, the free-form prediction is less good, especially for the predator population. [sent-152, score-0.194]

72 The approximate posterior displays nontrivial emerging properties: for example, we predict that there is a 10% chance that the prey population will become extinct at the end of the period of interest. [sent-154, score-0.286]

73 4 Example: gene autoregulatory network As a second example we consider a gene autoregulatory network. [sent-160, score-0.302]

74 The system consists again of two species, mRNA and protein; the process rates are given by fRN A (x + 1|x, y) = α (1 − 0. [sent-163, score-0.235]

75 99 × Θ (y − yc )) fRN A (x − 1|x, y) = βx fp (y + 1|x, y) = γx fp (y − 1|x, y) = δy (14) where Θ is the Heavyside step function, y the protein number and x the mRNA number. [sent-164, score-0.487]

76 The intuitive meaning of these equations is simple: both protein and mRNA decay exponentially. [sent-165, score-0.146]

77 On the other hand, mRNA production depends on protein concentration levels through a logical function: as soon as protein numbers increase beyond a certain critical parameter yc , mRNA production drops dramatically by a factor 100. [sent-167, score-0.641]

78 The dependence of the MF bound on the critical parameter yc is less straightforward and is given by T Lyc = const + 2 dt¯h (yc ) + log 1 − 0. [sent-172, score-0.349]

79 99 g 0 1 T T T h (yc ) dt 0 dt¯ g (15) 0 where g = gRN A (x + 1|x) qRN A and h (yc ) = y≥yc qp (y). [sent-173, score-0.103]

80 We can determine the minimum of (15) by searching over the possible (discrete) values of yc . [sent-175, score-0.317]

81 5 0 22 20 40 60 80 100 120 140 160 180 200 yc (a) (b) Figure 2: (a) Mean squared error (MSE) in the estimate of the parameters as a function of the number of observations N for the LV process. [sent-182, score-0.347]

82 (b) Negative variational likelihood bound for the gene autoregulatory network as a function of the critical parameter yc . [sent-183, score-0.576]

83 N 30 y 18 28 x 17 16 26 15 24 14 22 13 20 12 18 11 16 14 0 10 500 1000 1500 9 0 2000 t 500 1000 1500 2000 t (a) (b) Figure 3: MF approximation to posterior autoregulatory network process: (a) protein population and (b) mRNA population. [sent-184, score-0.393]

84 Again, we generated data by simulating the process with parameter values y c = 20, α = 2 × 10−3 , β = 6 × 10−5 , γ = 5 × 10−4 and δ = 7 × 10−5 . [sent-186, score-0.104]

85 Fifteen counts were generated for both mRNA and proteins, with initial count of 17 protein and 12 mRNA molecules. [sent-187, score-0.177]

86 The results of the approximate posterior inference are shown in Figure 3. [sent-189, score-0.14]

87 The inferred parameter values are in good agreement with the true values: yc = 19, α = 2. [sent-190, score-0.317]

88 Interestingly, if the data is such that the protein count never exceeds the critical parameter yc , this becomes unidentiﬁable (the likelihood bound is optimised by yc = ∞ or yc = 0), as may be expected. [sent-195, score-1.152]

89 5 Discussion In this contribution we have shown how a MF approximation can be used to perform posterior inference in MJPs from discretely observed noisy data. [sent-197, score-0.237]

90 While these focus on sampling from the distribution of reactions happening in a small interval in time, we compute an approximation to the probability distribution over possible paths of the system. [sent-200, score-0.146]

91 The factorisation assumption implies that the computational complexity grows linearly in the number of species D; it is unclear how MCMC would scale to larger systems. [sent-203, score-0.352]

92 An alternative suggestion, proposed in [11], was somehow to seek a middle way between a MJP and a deterministic, ODE based approach by approximating the MJP with a continuous stochastic process, i. [sent-204, score-0.122]

93 While these authors show that this approximation works reasonably well for inference purposes, it is worth pointing out that the population sizes in their experimental results were approximately one order of magnitude larger than in ours. [sent-207, score-0.185]

94 It is arguable that a diffusion approximation might be suitable for population sizes as low as a few hundreds, but it cannot be expected to be reasonable for population sizes of the order of 10. [sent-208, score-0.268]

95 The availability of a practical tool for statistical inference in MJPs opens a number of important possible developments for modelling. [sent-209, score-0.103]

96 It would be of interest, for example, to develop mixed models where one species with low counts interacts with another species with high counts that can be modelled using a deterministic or diffusion approximation. [sent-210, score-0.872]

97 All of these extensions rely centrally on the possibility of estimating posterior probabilities, and we expect that the availability of a practical tool for the inference task will be very useful to facilitate this. [sent-214, score-0.14]

98 Stochastic protein expression in individual cells at the single molecule level. [sent-221, score-0.098]

99 Bayesian inference for a discretely observed stochastic kinetic model. [sent-238, score-0.255]

100 Bayesian inference for stochastic kinetic models using a diffusion approximation. [sent-261, score-0.269]

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